3.12.74 \(\int \frac {2+x^4}{\sqrt [4]{-1+x^4} (-2+x^8)} \, dx\)
Optimal. Leaf size=86 \[ -\frac {1}{16} \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4+1\& ,\frac {-2 \text {$\#$1}^4 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 \log (x)+3 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-3 \log (x)}{\text {$\#$1}^5-\text {$\#$1}}\& \right ] \]
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Rubi [A] time = 0.27, antiderivative size = 171, normalized size of antiderivative = 1.99,
number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used
= {6725, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{58-41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[(2 + x^4)/((-1 + x^4)^(1/4)*(-2 + x^8)),x]
[Out]
-1/4*((58 - 41*Sqrt[2])^(1/4)*ArcTan[x/((2 - Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/Sqrt[2] - ((2 + Sqrt[2])^(5/4)
*ArcTan[x/((2 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/8 - ((58 - 41*Sqrt[2])^(1/4)*ArcTanh[x/((2 - Sqrt[2])^(1/4)
*(-1 + x^4)^(1/4))])/(4*Sqrt[2]) - ((2 + Sqrt[2])^(5/4)*ArcTanh[x/((2 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/8
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 6725
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx &=\int \left (-\frac {2+\sqrt {2}}{2 \sqrt {2} \left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}}+\frac {-2+\sqrt {2}}{2 \sqrt {2} \sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )} \, dx-\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {1}{2} \left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{58-41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [F] time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(2 + x^4)/((-1 + x^4)^(1/4)*(-2 + x^8)),x]
[Out]
Integrate[(2 + x^4)/((-1 + x^4)^(1/4)*(-2 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.29, size = 86, normalized size = 1.00 \begin {gather*} -\frac {1}{16} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(2 + x^4)/((-1 + x^4)^(1/4)*(-2 + x^8)),x]
[Out]
-1/16*RootSum[1 - 4*#1^4 + 2*#1^8 & , (-3*Log[x] + 3*Log[(-1 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-1
+ x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]
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fricas [B] time = 17.00, size = 1163, normalized size = 13.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)/(x^4-1)^(1/4)/(x^8-2),x, algorithm="fricas")
[Out]
1/8*sqrt(2)*(41*sqrt(2) + 58)^(1/4)*arctan(-1/98*(196*(17*x^5 - sqrt(2)*(12*x^5 - 17*x) - 24*x)*(x^4 - 1)^(3/4
)*sqrt(41*sqrt(2) + 58) + sqrt(2)*(2*(8*x^6 - 18*x^2 - sqrt(2)*(9*x^6 - 8*x^2))*sqrt(x^4 - 1) + (163*x^8 - 292
*x^4 - 2*sqrt(2)*(58*x^8 - 103*x^4 + 30) + 86)*sqrt(41*sqrt(2) + 58))*sqrt(-(782*sqrt(2) - 1107)*sqrt(41*sqrt(
2) + 58)) + 196*(3*x^7 - 4*x^3 - sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(41*sqrt(2) + 58)^(1/4)/(x^8 - 2))
+ 1/8*sqrt(2)*(-41*sqrt(2) + 58)^(1/4)*arctan(1/98*(sqrt(2)*(2*(8*x^6 - 18*x^2 + sqrt(2)*(9*x^6 - 8*x^2))*sqrt
(x^4 - 1) + (163*x^8 - 292*x^4 + 2*sqrt(2)*(58*x^8 - 103*x^4 + 30) + 86)*sqrt(-41*sqrt(2) + 58))*sqrt((782*sqr
t(2) + 1107)*sqrt(-41*sqrt(2) + 58))*(-41*sqrt(2) + 58)^(1/4) + 196*((17*x^5 + sqrt(2)*(12*x^5 - 17*x) - 24*x)
*(x^4 - 1)^(3/4)*sqrt(-41*sqrt(2) + 58) + (3*x^7 - 4*x^3 + sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(-41*sqrt
(2) + 58)^(1/4))/(x^8 - 2)) - 1/32*sqrt(2)*(-41*sqrt(2) + 58)^(1/4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 11*x) -
12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 + sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sqrt(2) +
58) + (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 + sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*sqrt(2)
+ 58) + sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(-41*sqrt(2) + 58)^(1/4))/(x^8 - 2)) + 1/32*sqrt(2)*(-41*sqrt(2
) + 58)^(1/4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 + sqrt(2)
*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sqrt(2) + 58) - (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 + sqr
t(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*sqrt(2) + 58) + sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(-41*sq
rt(2) + 58)^(1/4))/(x^8 - 2)) + 1/32*sqrt(2)*(41*sqrt(2) + 58)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 11*x) -
12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 - sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sqrt(2) +
58) + (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 - sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*sqrt(2)
+ 58) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(41*sqrt(2) + 58)^(1/4))/(x^8 - 2)) - 1/32*sqrt(2)*(41*sqrt(2) +
58)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 - sqrt(2)*(1
37*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sqrt(2) + 58) - (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 - sqrt(2)
*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*sqrt(2) + 58) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(41*sqrt(2)
+ 58)^(1/4))/(x^8 - 2))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 2}{{\left (x^{8} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)/(x^4-1)^(1/4)/(x^8-2),x, algorithm="giac")
[Out]
integrate((x^4 + 2)/((x^8 - 2)*(x^4 - 1)^(1/4)), x)
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maple [B] time = 23.20, size = 3206, normalized size = 37.28
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(3206\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4+2)/(x^4-1)^(1/4)/(x^8-2),x,method=_RETURNVERBOSE)
[Out]
-134217728/68921*ln(-(26388279066624*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf
(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(13
4217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^11*x^4-561030103040*RootOf(_Z^2-19480576*R
ootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*Root
Of(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272
*_Z^4+1)^7*x^4-687865856*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^5*RootOf(_Z^2-19480576*RootOf(1342
17728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(1342177
28*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*x^2-1951260475392*(x^4-1)^(1/4)*RootOf
(134217728*_Z^8-950272*_Z^4+1)^6*x^3+132070244352*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*RootOf(_Z^2-19480576*
RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)+2757033984*RootOf(_Z^2-1948
0576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+1948057
6*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-
950272*_Z^4+1)^3*x^4+2258403328*(x^4-1)^(3/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*x-2728960*(x^4-1)^(1/2)*R
ootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*Root
Of(134217728*_Z^8-950272*_Z^4+1)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134
217728*_Z^8-950272*_Z^4+1)^2)*x^2+13779789056*(x^4-1)^(1/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^2*x^3-1018363
904*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*
RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^2)-10820597*(x^4-1)^(3/4)*x)/(8192*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^
4-29*x^4-41))*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)
^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-
137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)+475136/68921*ln(-(26388279066624*RootOf(_Z^2-19480576*RootOf(13
4217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(13421
7728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)
^11*x^4-561030103040*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-
950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950
272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*x^4-687865856*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272
*_Z^4+1)^5*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^
4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1
)^2)*x^2-1951260475392*(x^4-1)^(1/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*x^3+132070244352*RootOf(134217728*
_Z^8-950272*_Z^4+1)^7*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8
-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-95
0272*_Z^4+1)^2)+2757033984*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728
*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z
^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*x^4+2258403328*(x^4-1)^(3/4)*RootOf(134217728*_Z^8
-950272*_Z^4+1)^4*x-2728960*(x^4-1)^(1/2)*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*R
ootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)*RootOf(_Z^2+19480576*RootOf(134217
728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*x^2+13779789056*(x^4-1)^(1/4)*RootOf(
134217728*_Z^8-950272*_Z^4+1)^2*x^3-1018363904*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*RootOf(_Z^2-19480576*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480576*RootOf
(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)-10820597*(x^4-1)^(3/4)*x)/(819
2*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4-41))*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*RootOf(_Z^2-19
480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(_Z^2+19480
576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)-1/41*RootOf(_Z^2-194
80576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*ln((2684354560*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^8*RootOf(_Z^2-19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6+137883*RootOf(
134217728*_Z^8-950272*_Z^4+1)^2)*x^4-687865856*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*RootOf(_Z^
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083008*(x^4-1)^(3/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*x+7599104*(x^4-1)^(1/2)*RootOf(_Z^2-19480576*RootO
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37883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2))/(8192*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4+41))+1/
41*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*
ln((-5905580032*RootOf(134217728*_Z^8-950272*_Z^4+1)^8*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+
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+1)^2)*x^2-5287968768*(x^4-1)^(1/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*x^3-38797312*RootOf(_Z^2+19480576*R
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272*_Z^4+1)^4*x^4+82624512*(x^4-1)^(3/4)*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*x-3148800*(x^4-1)^(1/2)*RootOf
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Of(134217728*_Z^8-950272*_Z^4+1)^4-3816*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*Roo
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1)^4-29*x^4+41))+RootOf(134217728*_Z^8-950272*_Z^4+1)*ln(-(2684354560*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)
^9-31641829376*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*x^2+1160773632*(x^4-1)^(1/4)*RootOf(134217
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4*RootOf(134217728*_Z^8-950272*_Z^4+1)*x^4-6437*(x^4-1)^(3/4)*x-87904*RootOf(134217728*_Z^8-950272*_Z^4+1))/(8
192*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4-41))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 2}{{\left (x^{8} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4+2)/(x^4-1)^(1/4)/(x^8-2),x, algorithm="maxima")
[Out]
integrate((x^4 + 2)/((x^8 - 2)*(x^4 - 1)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4+2}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4 + 2)/((x^4 - 1)^(1/4)*(x^8 - 2)),x)
[Out]
int((x^4 + 2)/((x^4 - 1)^(1/4)*(x^8 - 2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 2}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4+2)/(x**4-1)**(1/4)/(x**8-2),x)
[Out]
Integral((x**4 + 2)/(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x**8 - 2)), x)
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